On finite totally $2$-closed groups

Abdollahi, Alireza; Arezoomand, Majid; Tracey, Gareth

Comptes Rendus. Mathématique, Tome 360 (2022) no. G9, pp. 1001-1008.

An abstract group $G$ is called totally $2$-closed if $H=H^{\left(2\right),\Omega }$ for any set $\Omega$ with $G\cong H\le \mathrm{Sym}\left(\Omega \right)$, where $H^{\left(2\right),\Omega }$ is the largest subgroup of $\mathrm{Sym}\left(\Omega \right)$ whose orbits on $\Omega \times \Omega$ are the same orbits of $H$. In this paper, we classify the finite soluble totally $2$-closed groups. We also prove that the Fitting subgroup of a totally $2$-closed group is a totally $2$-closed group. Finally, we prove that a finite insoluble totally $2$-closed group $G$ of minimal order with non-trivial Fitting subgroup has shape $Z·X$, with $Z=Z\left(G\right)$ cyclic, and $X$ is a finite group with a unique minimal normal subgroup, which is nonabelian.

Classification : 20B05, 20D10, 20D25

[1] Abdollahi, Alireza; Arezoomand, Majid Finite nilpotent groups that coincide with their 2-closures in all of their faithful permutation representations, J. Algebra Appl., Volume 17 (2018) no. 4, 1850065, 11 pages | MR | Zbl

[2] Arezoomand, Majid; Abdollahi, Alireza; Spiga, Pablo On problems concerning fixed-point-free permutations and on the polycirculant conjecture-a survey, Trans. Comb., Volume 8 (2019) no. 1, pp. 15-40 | MR | Zbl

[3] Arezoomand, Majid; Iranmanesh, Mohammad A.; Praeger, Cheryl E.; Tracey, Gareth Totally $2$-closed finite groups with trivial Fitting subgroup (2021) (https://arxiv.org/abs/2111.02253)

[4] Aschbacher, Michael Finite Group Theory, Cambridge Studies in Advanced Mathematics, 10, Cambridge University Press, 1986 | Zbl

[5] Cameron, Peter J.; Giudici, Michael; Jones, Gareth A.; Kantor, William M.; Klin, Mikhail H.; Marušič, Dragan; Nowitz, Lewis A. Transitive permutation groups without semiregular subgroups, J. Lond. Math. Soc., Volume 66 (2002) no. 2, pp. 325-333 | DOI | MR | Zbl

[6] Dixon, John D.; Mortimer, Brian Permutation groups, Graduate Texts in Mathematics, 163, Springer, 1996 | DOI | Zbl

[7] Dobson, Edward; Kovács, István Automorphism groups of Cayley digraphs of ${\mathbb{Z}}_p^3$, Electron. J. Comb., Volume 16 (2009) no. 1, P149, 20 pages | MR | Zbl

[8] Doerk, Klaus; Hawkes, Trevor Finite soluble groups, De Gruyter Expositions in Mathematics, 4, Walter de Gruyter, 1992 | DOI | Zbl

[9] Evdokimov, Sergei; Ponomarenko, Ilya N. Two-closure of odd permutation group in polynomial time, Discrete Math., Volume 235 (2001) no. 1-3, pp. 221-232 | DOI | MR | Zbl

[10] Faradžev, Igor A.; Klin, Mikhail H.; Muzichuk, Mikhail E. Cellular rings and groups of automorphisms of graphs, Investigations in Algebraic Theory of Combinatorial Objects (Mathematics and its Applications), Volume 84, Kluwer Academic Publishers, 1994, pp. 1-152 | DOI | MR | Zbl

[11] Isaacs, I. Martin Finite group theory, Graduate Studies in Mathematics, 92, American Mathematical Society, 2008 | MR | Zbl

[12] Liebeck, Martin W.; Praeger, Cheryl E.; Saxl, Jan A classification of maximal subgroups of the finite alternating and symmetric groups, J. Algebra, Volume 111 (1987) no. 2, pp. 365-383 | DOI | MR | Zbl

[13] Liebeck, Martin W.; Praeger, Cheryl E.; Saxl, Jan On the $2$-closures of finite permutation groups, J. Lond. Math. Soc., Volume 37 (1988) no. 2, pp. 241-252 | DOI | MR | Zbl

[14] Mathoverflow 2-closure of a permutation group, 2016 (http://mathoverflow.net/questions/235114/2-closure-of-a-permutation-group)

[15] Monks, Kenneth M. The mobius number of the symmetric groups, Ph. D. Thesis, Colorado State University, USA (2012)

[16] O’Nan, Michael E. Estimation of Sylow subgroups in primitive permutation groups, Math. Z., Volume 147 (1976), pp. 101-111 | MR | Zbl

[17] Ponomarenko, Ilya N. Graph isomorphism problem and $2$-closed permutation groups, Appl. Algebra Eng. Commun. Comput., Volume 5 (1994) no. 1, pp. 9-22 | DOI | MR | Zbl

[18] Ponomarenko, Ilya N.; Vasil’ev, Andreĭ V. Two-closure of supersolvable permutation group in polynomial time, Comput. Complexity, Volume 29 (2020) no. 5, 5, 33 pages | DOI | MR | Zbl

[19] Praeger, Cheryl E. On elements of prime order in primitive permutation groups, J. Algebra, Volume 60 (1979), pp. 126-157 | DOI | MR | Zbl

[20] Praeger, Cheryl E.; Saxl, Jan Closures of finite primitive permutation groups, Bull. Lond. Math. Soc., Volume 24 (1992) no. 3, pp. 251-258 | DOI | MR | Zbl

[21] Vasil’ev, Andreĭ V.; Churikov, Dmitriy V. $2$-closures of $\frac{3}{2}$-transitive groups in polynomial time, Sib. Math. J., Volume 60 (2019) no. 2, pp. 279-290 | DOI | MR | Zbl

[22] Wielandt, Helmut Permutation groups through invariant relations and invariant functions, Volume 1 Group Theory, Part 1 (Huppert, Bertram et al., eds.) (Mathematische Werke / Mathematical Works), Volume 1, Walter de Gruyter, 1994, pp. 237-296 also published in Lectures Notes, Ohio State University, (1969) | DOI | Zbl

[23] Xu, Jing Metacirculant tournaments whose order is a product of two distinct primes, Discrete Math., Volume 311 (2011) no. 8-9, pp. 571-576 | MR | Zbl

[24] Xu, Jing Digraph representations of 2-closed permutation groups with a normal regular cyclic subgroup, Electron. J. Comb., Volume 22 (2015) no. 4, P4.31, 14 pages | MR | Zbl

[25] Xu, Jing; Giudici, Michael; Li, Cai Heng; Praeger, Cheryl E. Invariant relations and Aschbacher classes of finite linear groups, Electron. J. Comb., Volume 18 (2011) no. 1, P225, 33 pages | MR | Zbl